X₁₆₃

TRILINEAR PRODUCT X₆*X₁₁₀

X₁₆₃ = barycentric product of circumcircle intercepts of line X₁X₂₁

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₆₃ (TRILINEAR PRODUCT X₆*X₁₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {101, 110}, {109, 112}

• Related Centers: X₁₅₇₇ (ISOGONAL CONJUGATE OF X₁₆₃), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT))

Trilinears: \((\sin 2A)/(\sin 2B - \sin 2C) : (\sin 2B)/(\sin 2C - \sin 2A) : (\sin 2C)/(\sin 2A - \sin 2B)\)

Barycentrics: \(a^{3}/(b^{2} - c^{2}) : b^{3}/(c^{2} - a^{2}) : c^{3}/(a^{2} - b^{2})\)

Search Value (6, 9, 13): \(0.422785826392\)

▸ Alternate Coordinates1 form

• Trilinears: \(a^{2}/(b^{2} - c^{2}) : :\)

▸ Incident Central Lines13 lines

{1, 293}, {19, 563}, {31, 923}, {32, 849}, {48, 1094}, {99, 825}, {101, 110}, {109, 112}, {284, 909}, {643, 1018}, {692, 906}, {798, 1101}, {813, 827}

▸ Functional Relations15 relations

• X₁₆₃ = isogonal conjugate of X₁₅₇₇

• X₁₆₃ = isotomic conjugate of X₂₀₉₄₈

• X₁₆₃ = complement of X₂₁₂₉₄

• X₁₆₃ = anticomplement of X₂₁₂₅₃

• X₁₆₃ = barycentric product of circumcircle intercepts of line X₁X₂₁

• X₁₆₃ = crosssum of X₆₅₆ and X₆₆₁

• X₁₆₃ = X_i-aleph conjugate of X_j for these (i,j): (648, 19), (662, 610)

• X₁₆₃ = trilinear product of PU(2)

• X₁₆₃ = barycentric product of PU(70)

• X₁₆₃ = trilinear product of X₅₈X₁₀₁

• X₁₆₃ = trilinear product of the 6 vertices of the 1st and 2nd circumperp triangles

• X₁₆₃ = trilinear pole of line X₃₁X₄₈

• X₁₆₃ = X₉₂-isoconjugate of X₆₅₆

• X₁₆₃ = crossdifference of every pair of points on line X₁₁₀₉X₂₆₃₂

• X₁₆₃ = trilinear product X₆*X₁₁₀

Notes

Let A'B'C' be the circumcevian triangle of X₅₁₂. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. A″, B″, C″ are collinear on line X₆₆₇X₇₈₈. The lines AA″, BB″, CC″ concur in X₁₆₃. (Randy Hutson, August 19, 2019)